Basics of trigonometryToday we will study trigonometric ratios of 90°, 180°, and 270°.
5) Trigonometric ratios of an angle O = 90°
Here angle AOP is 90°.
According to the diagram, OP = r, and the coordinates of point P are (0, r). So x-coordinate of point P is 0 and the y-coordinate of point P is r.
So, all the trigonometric ratios of O = 90° with
x = 0, y = r, r = r.
a) sin 90° = y/r
sin 90° = r/r
sin 90° = 1
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b) cos 90° = x/r
cos 90° = 0/r
cos 90° = 0
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c) tan 90° = y/x
tan 90° = r/0
tan 90° = ∞
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d) csc 90° = r/y
csc 90° = r/r
csc 90° = 1
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e) sec 90° = r/x
sec 90° = r/0 sec 90° = ∞
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f) cot 90° = x/y
cot 90° = 0/r
cot 90° = 0
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6) Trigonometric ratios of an angle O = 180°
Here angle AOP is 180°.
According to the diagram, OP = r, and the coordinates of point P are (-r, 0). So the x-coordinate of point P is -r and the y-coordinate of point P are 0.
So, all the trigonometric ratios of O = 180° with
x = -r, y = 0, r = r.
a) sin 180° = y/r
sin 180° = 0/r sin 180° = 0
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b) cos 180° = x/r
cos 180° = -r/r
cos 180° = -1
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c) tan 180° = y/x
tan 180° = 0/-r tan 180° = 0
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d) csc 180° = r/y
csc 180° = r/0
csc 180° = ∞
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e) sec 180° = r/x
sec 180° = r/-r sec 180° = -1
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f) cot 180° = x/y
cot 180° = -r/0
cot 180° = -∞
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7) Trigonometric ratios of an angle O = 270°
Here angle AOP is 270°.
According to the diagram, OP = r, and the coordinates of point P are (0, -r). So x-coordinate of point P is 0 and the y-coordinate of point P is -r.
x = 0, y = -r, r = r.
a) sin 270° = y/r
sin 270° = -r/r
sin 270° = -1
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b) cos 270° = x/r
cos 270° = 0/r
cos 270° = 0
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c) tan 270° = y/x
tan 270° = -r/0
tan 270° = ∞
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d) csc 270° = r/y
csc 270° = r/-r
csc 270° = -1
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e) sec 270° = r/x
sec 270° = r/0
sec 270° = ∞
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f) cot 270° = x/y
cot 270° = 0/-r
cot 270° = 0
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Actually, it is very easy to understand the concept once we found out the values of x, y, and r for the point P on the terminal ray.
In the next Blog, we will study all the trigonometric ratios for the following groups.
Group-01: 120° (90° + 30°), 210° (180° + 30°), 300° (270° + 30°).
Group-02: 135° (90° + 45°), 225° (180° + 45°), 315° (270° + 45°).
Group-03: 150° (90° + 60°), 240° (180° + 60°), 330° (270° + 60°).
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